Graphing Software in Understanding of Polynomial …sms35077/eed602/portfolio/research... · Web viewIn this study statistical analysis was done by using SPSS computer program. The

Graphing Software 1

Graphing Software in Understanding of Polynomial Functions

Sadia M Syed

California State University, Northridge

College of Education

06/04/2008

Instructor, Dr. Brian Foley

Graphing Software 2

TABLE OF CONTENTS

Abstract

Introduction

o Description of the Study

o Importance of the study

o The Research Questions

o Description of the Tool

Literature Review

o Theoretical Framework

o Related Studies

Methodology

o Description of the Participants

o Description of the Tool

o Description of how the Tool was used

o Research Instruments

o Procedure

o Analysis

Findings

o Findings from post-test

o Findings from student survey

Conclusion

References

Appendix A – Student Survey 1

Appendix B – Post-Test

Appendix C – Student Survey 2 - Experimental Group

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ABSTRACT

This research was conducted to find out whether or not using the

software, Graph (www.padowan.dk/graph/), during the instruction of

polynomials can improve students’ understanding of function and their

ability to create and interpret graphical representation of functions. This

study was conducted over three weeks and with two groups (control

group and experimental group) of students. Both groups were taught

by the same teacher and were instructed the same lesson on

polynomial function. After each lesson the control group was asked to

work with graph-paper and the experimental group was asked to work

with the software. Both groups were assessed by post-test, and a

student survey was conducted after the post-test with the experimental

group to gather information regarding students’ opinion on the software

and its influence on their learning process. The t-test analysis of the

post-test showed that there is no overall significant difference between

the understanding of the concepts of the control group and the

experimental group. But the software helped the students of

experimental group as they did well on the question of analyzing and

interpreting graphs. Moreover, the analysis of the ‘student survey’

conducted with the experimental group indicated that using software

during the instruction improved students’ motivation, confidence and

interest.

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INTRODUCTION

Description of the Study:

In last several years functions and graphs have been a major focus of many

researches in the field of mathematics and education. Several studies have been

conducted to understand the influence of computers on students’ understanding of the

concept of functions, and their ability to create and interpret graphical representation of

functions (e.g., Asp, Dowsey & Stacey, 1994; Hollar & Norwood, 1999; Kaput, 1992;

Manoucherhri, 1999; Ruthven & Hennessy, 2002; Simmt, 1997; Yerushalmy, 1991).

The findings from these studies have provided strong indication that the use of

computers as a thinking aid and an intellectual tool enrich learners’ mathematical

understanding, facilitate students’ growth of mathematical explorations, and improve

their problem solving skills and concept developments.

Most secondary schools in the USA are now equipped with computers and

connected to the internet. According to Educational Technology Fact Sheet (2006), the

ratio of students to computers in all public schools in 2003 was 4.4 to 1. The

Mathematics 2000 report states that the availability of computers in classrooms

increased by at least 20 percentage points from 1996 to 2000, although the use of

computers in mathematics teaching increased at a lower rate than other subjects

(Paulson, 2000).

Technology provides students an opportunity to use "hands-on" techniques in

problem solving. Technology also helps students to develop an understanding of the

processes and reasoning that are the heart of mathematical problem solving (Hudnutt,

2007). The NCTM also supports the use of technology to enhance student learning. As

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stated in one of the seven principles in the Principles and Standards, “Calculators and

computers are reshaping the mathematical landscape, and school mathematics should

reflect those changes. Students can learn more mathematics more deeply with the

appropriate and responsible use of technology. They can make and test conjectures.

They can work at higher levels of generalization or abstraction,” (NCTM, 2000, p. 25).

This research was conducted to find out whether or not using the software

(Graph) during the instruction of polynomials can improve students’ understanding of

function and their ability to create and interpret graphical representation of functions.

The software that will be used in this study is ‘Graph’, a free graphing software from

www.padowan.dk/graph/. The software can be used to draw mathematical graphs in

various (i.e. Cartesian, Polar) coordinate systems. Users can easily draw graphs of

functions and the program makes it very easy to visualize the functions. Students can

readily enter a list of expressions using the graph editor, watch the graph, and explore a

function through a numeric display of coordinates, intersection points, slopes, tangents

and maxima. All these features are designed to be accessed through buttons and

‘Menu’ commands, and are extremely powerful for users of mathematics who are

seeking data on a specific function or equation. I have chosen this software rather than

other advanced graphing software, because it is easy to understand and manipulate

(see the screen shots in chapter 3) by algebra beginners.

One of the important reasons for choosing this software is its ease of

manipulation of the command bar. According to Yerushalmy (1999), one of the main

problems of past and present graphing software is their complex manipulation and lack

of user-friendly features. Using very simple commands this software allows students to

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easily enter a list of functions and explore those through various displays of coordinates,

intersection points, slopes, tangents and maxima. Another problem identified by Guin

and Trouche (1999) is the distinction between the function syntax of one set of

commands and the solving syntax of the other commands that initiated difficulties for

students who are at the early stage of developing conceptual understanding of

functions. The software ‘Graph’ also meets the current focus of mathematics curriculum,

which emphasize that the school should provide opportunities for students to construct

knowledge and think mathematically through exploration and investigation

(Manoucherhri, 1999).

Technology can have profound effect in the learning and teaching of functions

and graphs. In their analysis of research on the teaching and learning of functions,

Leinhardt, Zaslavsky, and Stein (1990) note that, “more than perhaps any other early

mathematics topic, technology dramatically affects the teaching and learning of

functions and graphs,” (p. 7). With the help of technology teachers can have students

make observations and conjectures within a variety of function representations such as

equations, graphs, and tables. According to Hudnutt (2007) “Students can then begin to

make connections among the representations in order to develop a concept image

without first having an in depth knowledge of function. With the use of technology,

teachers can expose students at a much earlier stage in their cognitive development to

the function concept. This, in turn, allows students to explore the connections among

representations enabling the learning of functions to become investigative in nature.”

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Importance of the Study:

Recent reform movements in mathematics education encourage the use of

computer technologies in the classroom, and in particular, the use of computer

supported explorations as contexts for mathematics instructions (Manoucherhri, 1999).

The availability of computers and mathematics software (Freeware, Shareware etc.) has

great potential to take a positive step towards engaging students more actively in a

process of mathematical thinking and learning. Even though all aspects of a complex

mathematical idea can not be expressed with a single representational system, Kaput

(1992) argues that the ability to make translation from one representation of a function

to another is a particularly important aspect of mathematical thinking which may be

enhanced by technology. The convenient access provided by graphing software to

numerical and graphical representations of a verity of functions may assist students to

develop a broader and deeper understating of the concepts. Graphing software enables

rapid and automatic translation between algebraic, graphical and numerical

representations, whereas translation by hand is generally a slow and laborious process

for students.

The study of polynomials of higher degree can become a fascinating part of

school mathematics with the accessibility to graphical representations now available

through computer software and graphing calculators. These representations can be

incorporated and used to create a mental image of the functions as an aid to

mathematical intuition needed to deal with functions (Movshovitz-Hadar, 1993). Since

"seeing" the algebra has become possible through graphing technology, it can be used

to make symbol manipulations more meaningful in operating the algebra.

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Understanding the concepts of polynomial functions and their graphs using

graphing software will provide students with insights that enable them to construct

deeper and more coherent graphing concepts and it will help them develop important

techniques needed to comprehend various concepts of polynomial functions. For

example, when the students are learning translation and transformation of graphs, they

can be asked to use the software to graph functions with different degrees and with

different coefficients. Using this software the students can easily examine and discover

how change in degree and coefficient of a function can change its shape and position.

Following are some screen shot that display some of the transformations and

translations created by the software.

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Figure 1

Graphing software provide more emphasis on graphs and their interpretation,

both to help students understand key ideas of polynomial functions, their transformation

and translation. According to the discussion of Kissane (1995), the ease with which

calculators can draw graphs means that students can concentrate on the meanings

inherent in graphs instead of the mechanics of producing them. Using graphing software

in learning of polynomial concepts gives more importance on making the ‘use of’

graphs, rather than ‘producing’ a graph.

Another profound reason for conducting this study on effectiveness of graphing

software in learning of polynomial concepts is to encourage teachers and students of

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mathematics education to take advantage of vast collection of freeware and shareware

available on the internet. Schools in lower economic status can extend their instructional

facility by combining their existing computer technologies with the graphing software.

Many schools may not be able to provide enough graphing calculators for all students,

but can probably use their computers equipped with this free graphing software, and

thus create a significant connection between uses of computer and mathematics

education. Through this research I think other researchers will be encouraged to study

and evaluate the extensive collection of educational freeware and shareware available

on the internet and also be able to make recommendations on their applications by

using them with traditional instructions.

The Research Questions:

This study was conducted over three weeks and with two groups of students.

Both groups were taught by the same teacher and they were equivalent in their

academic background and knowledge in understanding functions. These groups were

instructed the same lesson on polynomial function. After each lesson one group was

asked to work with graph-paper and the other group was asked to work with the

software. The research question of this study was to find out whether or not using the

software (Graph) during the instruction of polynomials can improve students’

understanding of function and their ability to create and interpret graphical

representation of functions. This question will be answered through the analysis of the

students’ (both groups) score on their post-test.


LITERATURE REVIEW

Theoretical Framework:

Use of graphing technology in mathematics instructions is strongly influenced by

Jean Piaget’s “Constructivism.” The perspective of constructivism on teaching and

learning of mathematics “focuses on the provision of opportunities for students to

engage in reflective mathematical thinking as they consider the viability of their extant

understandings, strive to resolve creatively new cognitive perturbations, and test the

viability of their tentative solutions strategies” (Forster & Taylor, 2000, p. 4). In the article

“Calculators and Constructivism” by Wheatley and Clements (1990), it is cited from

Glasersfeld (1990) that from a constructivist perspective, a calculator can aid

mathematics learning when it “permits meaning to be the focus of attention, creates

problematic situations, facilitates problem solving, allows the learner to consider more

complex tasks, and lends motivation and boosts confidence” (p. 296). Graphing

technology creates a constructive environment that helps students to explore

mathematical concepts in an organized way. According to Yerushalmy (1999), graphing

software helps to organize mathematical concepts by a limited collection of important

terms, objects and actions. Graphing technology makes these organizations visible to

user and thus becomes a strong mathematical thinking tool, tool for planning, and tool

for problem posing.

The inquiry of using the graphing technology in mathematics instructions is also

influenced by the socio-cultural view, described by Forster & Taylor (2000). According

to Forster & Taylor, socio-cultural perspectives focus on the provision of opportunities

for students to co-construct valid mathematical meaning through participation in rich


mathematical conversations with the teachers, graphing calculators and fellow students.

In their research with graphing calculators, Forster & Taylor (2000) found most of the

students who used calculator during the instructions were able to successfully transfer

their knowledge from a specific situation (without calculator) to another situation when

they were permitted to use calculator. They described the transfer of knowledge of

those students by Cobb and Bower’s (1999) social constructivist view, which explains

that the use of graphing technology is social in a sense that students enter an

interactive, intellectual partnership with it. From this perspective, the transfer of

knowledge represents that students’ mathematics practices in one social context (an

instructional setting without technology) were relevant to another social context (where

instruction extended to using technology). From this research it is noticeable that when

students experiment various concepts using graphing calculator, it can be directed as a

transferable social practice of technology usage.

Research on Graphing Software:

A significant number of studies (Asp, Dowsey & Stacey, 1994; Hollar & Norwood,

1999; Kaput, 1992; Manoucherhri, 1999; Ruthven & Hennessy, 2002; and Simmt, 1997)

have been conducted to examine the effects of graphing technologies on students

understanding of algebraic functions. By generalizing the findings of the research, it is

evident that students’ achievement is positively influenced by the graphing technologies

when they are used to facilitate students’ higher cognitive skills (i.e. analyze,

synthesize, evaluate). A number of studies (Heller & Curtis, 2006; Khoju & Miller, 2005;

Ellington, 2003) show that the use of graphing technology positively impacts students’


performance in algebra, improves mathematics test scores – both with and without a

calculator during testing, and establishes better student attitudes towards math.

Graphing technology helps increase students’ problem solving skills. According

to Dunham & Dick (1994) the use of graphing technology when teaching problem

solving strategies led to a significant increase in the achievement of the students.

Graphing technology facilitates students to create relationships to new situations and to

communicate solution strategies (Hubbard, 1998). Graphing technology lessens the

amount of attention needed for algebraic manipulation, thus allowing more time for

actual instruction. It also supplies more functions and can serve as a monitoring aid

during the problem solving process (Dunham & Dick, 1994).

A meta-analysis of eight individual studies done by Khoju & Miller (2005) has

found strong evidence of increased performance in algebra when students increasingly

use graphing technology. On the other hand, Barton (2000) has expressed that when

studies seeking to isolate the technology variable, by controlling curriculum, texts,

homework, exams and teacher variables, did not find a significant difference in overall

achievement between the treatment group and the control group. Barton (2000) also

suggests that simply having access to technology does not ensure it will be used to

enhance learning. A meta-analysis of 54 studies done by Ellington (2003) found that

students who receive instruction using a graphing calculator perform as well or

significantly better in conceptual problem solving and operational skills areas. One of

the studies conducted by National Assessment of Education Progress (NAEP) has

shown that frequent use of graphing technology is associated with greater mathematics

achievement:


Eighth-graders whose teachers reported that calculators were used almost everyday scored highest. Weekly use was also associated with higher average scores than less frequent use. In addition, teachers who permitted unrestricted use of calculators and those who permitted calculator use on tests had eighth-graders with higher average scores than did teachers who did not indicate such use of calculators in their classrooms (National Center for Education Statistics 2001, p. 141).

Graphing technology can make designing, creating and using multiple

representations easier. When using this technology during instruction, rather than using

time for laborious and tedious calculations, students can have more time and mental

energy to explore various underlying concepts. As an example, during instruction on

quadratic equations, students can easily investigate the effects of changing the value of

a, b and c on the graph of ax2 + bx + c, which can be very tedious when using paper-

pencil graphing techniques. Specific research has shown that students can often reason

best when they experience mathematics through related representation, such as

equations, tables and graphs (Goldenberg, 1995; Kaput, 1992). Furthermore,

technology can create connection between the representations, enabling students to

make conceptual connections, such as, understating how a change in an equation links

to a change in a graph (Roschelle, 2006).

In regards to the ease in which functions can be graphed and manipulated with

software, Dugdale (1993) found that these tools have, “raised the possibility of visual

representations of functions playing a more important role in mathematical reasoning,

investigation, and argument. Relationships among functions can be readily observed,

conjectures can be made and tested, and reasoning can be refined through graphical

investigation,” (pg. 115). Doerr and Zangor (2000) analyzed a pre-calculus class that


widely used the graphing calculator. They made in-depth observations and found that

this technology was instrumental in facilitating analytical thinking.

Tall (1989) examined the potential of well-designed software to improve a

learner’s concept image of function by allowing the learner to explore the complex

structures of functions. The software allows students to experience higher level

cognitive structures than they would be able to without such software. He suggested

students can more readily explore complex concepts through the use of technology.

Dugdale (1993) also agrees with Tall. In her own research on the use of technology to

support student thinking with functions, as well as reviewing others’ research, Dugdale

states, “Such tools have facilitated the movement away from a focus on calculating

values and plotting points toward a more global emphasis on the behavior of entire

functions, end even families of functions,” (pg 114). Therefore, using graphing software

students can study the more complex global aspects of function prior to or in parallel

with studying functions as input/output machines.

Graphing technology helps to improve students’ performance on visual and

graphing tasks. According to Demana & Waits (1992), the visual impact of graphing

calculators on students greatly enhances their learning of mathematics. Because

graphing technologies enhance visualization and invite self discovery, students are able

to relate to novel problem situations (Scariano & Calzaada, 1994, cited by Hubbard,

1998). Graphing technology also serves as a positive motivator among students

because they seem to enjoy using it. In their research, Demana & Waits (1992) have

shown that graphing calculators can make the study of mathematics fun and can give

students excellent learning experiences.


Using dynamic software students can be benefited by the easy manipulation of a

function through various representations to construct their own internal, flexible images

of function. Yerushalmy and Chazan (1990) in their research with dynamic geometry

environment found that students were able to reason more flexibly about geometric

concepts than their counterparts who learned with static diagrams. Similarly,

Moschkovich, Schoenfeld, and Arcavi (1993) explored student learning of functions with

the dynamic software, GRAPHER, and found that such an environment “allows students

to operate on equations and graphs as objects… in ways not possible before the

existence of such technologies,” (pg. 98). In this research study, they also found that “it

is not just the dynamics on the screen that makes the difference in student learning.

Students must integrate what they see on the computer screen into their own

conceptual structures for any learning to take place” (Moschkovich et al,1993).

A number of studies (Heller & Curtis, 2006; Khoju & Miller, 2005; Ellington, 2003)

have shown that using graphing technology in the instruction of algebra improves

students’ assessment scores and positively impacts students’ performance in algebra.

Heller & Curtis (2006) conducted a study to look at the relationship between graphing

calculator use and student standardized test scores in grades 9-11. One of the key

findings of their study was increasing use of graphing calculators during instruction

resulted in higher test scores even when students did not use graphing calculators

during test taking. Ruthven (1990) in his study suggested the impact of technology in

the secondary classroom might depend on the way in which the technology is used to

mediate mathematics in the classroom.


Use of graphing technology in the classroom can be different depending on the

curriculum and instruction. Simmt (1997) in the study “Graphing Calculators in High

School Mathematics,” examines how mathematics educators used graphing calculators

in their instructions and how their views of mathematics were manifested in the ways

they choose to use this technology. In the study Simmt (1997) reported that ‘most of the

teachers used the tool to facilitate one or two guided-discovery activities, and they did

not use the tool to facilitate and/or encourage the students to conjecture and prove or

refute ideas’ (p. 286). In the article “Graphing Calculators in the Mathematics

Classroom” Smith (1998) stated, “When wearisome computation and plotting tasks are

minimized, students can become engaged in answering "what-if" questions. The

thought of changing the premises of a mathematical argument grows more attractive if

the chore of executing those changes is easier. Moreover, the graphing calculator

promotes autonomy in asking questions, encouraging students to pose their own

problems (p. 1).” Touval (1997), in the research "Investigating a Definite Integral - From

Graphing Calculator to Rigorous Proof," suggests that the graphing calculator can be

used as a springboard for discovery. While learning how to calculate definite integrals,

students proposed their own theories concerning integration. While using the graphing

calculators to investigate quick solutions to problems, the students formulated

conjectures that eventually led to a rigorous proof.

Teachers’ level of expertise in using graphing software has an impact on the

proper implementation of the technology in the classroom, and teacher’s level of

expertise has to be increased by proper training before conducting the research. Asp,

Dowsey and Stacey (1994) conducted a study using ANUGraph software package to


teach linear and quadratic functions. Their finding suggested teachers’ previous

experiences of using graphing technology have a significant impact on successful

implementation of these tools in mathematics instructions. One of the key findings of the

research done by Heller & Curtis (2006) described that when the teachers participated

in training on how to use graphing calculator or other computerized graphing

technology, student achievement was significantly higher, compared to those teachers

who learned to use those technology by reading manual. This finding suggests that

students benefit when their teachers receive professional development that is specific to

graphing technology in math instruction.

Even though there are many studies that deal with the positive impacts of

graphing software and calculators, there are few studies that have found some

shortcomings. Goldenberg (1988) identified student difficulties with graphing

technologies due to issues of scale. Students unfamiliar with notions of window sizing,

scaling on the axes, and global behaviors experienced difficulties in using the

technology to understand graphs. When computers are used for simulation of

mathematical constructs, there are still opportunities for misinterpretation. Moschkovich,

Schoenfeld, and Arcavi (1993) found students interpreted the pixilation of a line

represented on a graph as an actual property of the line.


METHODOLOGY

Description of the Participants:

The participants in this study were middle and high school students. The school

is located in Eagle Rock, a suburb of Los Angeles. There were total thirty students in

the study, and each group contained fifteen students. Among the participating students

seventeen (57%) were female participants and thirteen (43%) were male participants;

nineteen (63%) participants were Hispanic, seven (23%) were Caucasian, two (7%)

were African American, and two (7%) were Asian. Approximately 34% of the

participants were English Language Learners (ELL) and 47% of the participants were

classified as economically disadvantaged. Among the participants there were twelve

(40%) eighth grade students, fifteen (50%) ninth grade students and three (10%) tenth

grade students. Approximately 85% of these participants had access to personal

computer at home and 75% had Internet connection at home. According to a survey

(See Appendix A), majority of the participants used their computers mostly to browse

the internet, send and receive email and text massage, and perform word processing

tasks. The participants had one 45 minutes math period everyday from Monday through

Friday.

‘Matching’ process was used to create relatively similar “Control Group” and

“Experimental Group” for the research, and each group had fifteen participants. The

reason for using ‘Matching’ process for this study was to create two groups that

contained participants who had similar scores in the Algebra Diagnostic Test. In this

school, at the beginning of each academic year every algebra student takes a

diagnostic test on algebra to determine their previous understanding about the


prerequisite concepts. Based on the test scores, then the students are grouped for the

respective level/class on algebra. The researcher had access to those diagnostic test

scores of the participating students. Students in all levels of scores (i.e. participants who

had more than 90%, had less than 90% and more than 80%) were equally distributed

between the two groups (Control and Experimental). Ten participants had more than

90% on that test and they were equally divided between the two groups. Ten

participants had scores between 90% and 81% on that test and they were also equally

divided between the two groups. Ten participants had scores between 80% and 70%

and they were also equally divided between the two groups.

Materials:

Description of the Tool:

The software that will be used for this study is ‘Graph’ (www.padowan.dk/graph/).

This software can be used to draw mathematical graphs in various (i.e. Cartesian,

Polar) coordinate systems. Students can readily enter a list of expressions using the

graph editor, watch the graph, and explore a function through a numeric display of

coordinates, intersection points, slopes, tangents and maxima. All these features are

designed to be accessed through buttons and ‘Menu’ commands. According to the

software publisher, this application allows students to do the following:

Draw functions: Graph can draw normal functions, parameter functions, and

polar functions. Students can use a lot of built-in functions, e.g. sin, cos, log, etc.

Students may specify color, width and line style of the graphs and the graphs may me

limited to an interval. It is also possible to show a circle at the ends indicating open or

closed interval.


Screen Shot # 1

Draw relation: Graph can show any equation and inequality, for example sin(x)

< cos(y) or x^2 + y^2 = 25. Students can choose line width and color for the equations,

and color and shading style for the inequalities.

Interact with other programs: Students can save the coordinate system with

graphs as an image on disk either as a bitmap (bmp), Potable Network Graphics (png),

JPEG, metafile (emf) or Portable Document Format (PDF).

Evaluate: Given an x-coordinate the software will calculate the function value

and the first two derivatives for any given function. Alternatively the function may be

traced with the mouse.

Calculate: Graph can help students calculate the area under a function in a

given interval and the distance along the curve between two points on the function. The


program can also show the first derivative of a function, and create tangents and normal

lines to a function at given coordinates.

Screen Shot # 2

Point series and trend lines: Students can create series of points with different

markers, colors and size. Data for a point series can be imported from other programs,

e.g. Microsoft Excel. It is possible to create a line of best fit from the data in a point

series, either from one of the built-in models or from a user specified model.

Shadings and labels: Graph can insert shadings used to mark an area related

to a function. Shadings may be created with different styles and colors in a user

specified interval.

Description of how the software used in the research:

The software that was used for this study is ‘Graph’ (www.padowan.dk/graph/).

Using the software students can easily draw graphs of functions and the program


makes it very easy to visualize the functions. Students can readily enter a list of

expressions using the graph editor, watch the graph, and explore a function through a

numeric display of coordinates, intersection points, slopes, tangents and maxima.

Students can also use the images of the graphs in other programs by saving it as a

bitmap (bmp), Potable Network Graphics (png), JPEG, metafile (emf) or Portable

Document Format (PDF). All these features are designed to be accessed through

buttons and ‘Menu’ commands.

The first concept that was introduced for this unit of instruction is ‘Polynomial

Models’ where students learn the definition of polynomial as well as various types of

polynomials (i.e. quadratic polynomials, cubic polynomials) that can arise in real life

situation. During the lesson teacher used direct instruction at the beginning to clarify

various vocabularies (i.e. degree, leading co efficient, quadratic and cubic polynomials,

factoring, zeros of polynomial). In the second part of the lesson teacher used the

software to demonstrate some graphs of polynomials. Simultaneously teacher

demonstrated how to insert functions and create graphs using the software. Towards

the end of this lesson students worked in groups of two in inserting functions and

creating graphs using the software. Students were given a worksheet with various

equations. Among those equations some were polynomial and some were exponential

functions. Students were asked to create graphs of those equations and they were also

encouraged to conjecture regarding the type of functions. At the end of the lesson

students discussed and provided support for their conjectures.

The second concept was ‘Transformation and Translation of Quadratic and Cubic

Polynomial’ where student learn how the graphs of polynomial functions change their


size, shape, and position in respect to change in their degrees and coefficients. In this

lesson teacher used guided discovery based instructions to encourage students explore

‘what-if’ types of questions. For example, to understand translation and transformation

of quadratic graphs, teacher designed a set of functions for students to explore using

the software. In one of the specific assignments, teacher asked the following: A) Use all

integer values for ‘a’ where -10<a<10 in the given function f(x) = ax2 to create graphs.

What can you conclude about the effect of the value of ‘a’? B) Use all integer values for

‘b’ where -10<b<10 in the given function f(x) = x2+bx to create graphs. What can you

conclude about the effect of the value of ‘b’? C) Use all integer values for ‘c’ where -

10<c<10 in the given function f(x) = x2+x+c to create graphs. What can you conclude

about the effect of the value of ‘c’? For this assignment students worked in groups of

two where they used the software to explore various changes in the graphs and

developed their understanding. The software created a dynamic environment that

helped the students to observe the translations and transformations of the graphs.

The third concept that was discussed in this unit was ‘Factoring Polynomials’

where students learned four common ways of factoring: 1) factoring the largest common

monomial factor, 2) factoring following a pattern, 3) quadratic polynomial factoring, 4)

using the Factor Theorem. This lesson was delivered using direct instruction and group

discussions. Students worked individually to work out exercises that were presented

through worksheets. The use of the software was minimal during this lesson.

The forth concept was ‘Factor Theorem and Identifying Zeros of the Polynomial

Function’ where students learned the Factor Theorem to identify the zeros of a

polynomial function. In this lesson teacher designed the instructional tasks involving a


various degrees of polynomials. In these tasks student were required to use the

software to create the graphs of those given polynomials and by analyzing the x-

intercepts of those graphs they further developed the understanding of relations

between zeros of the polynomials. For example, after analyzing the graphs student

would recognize and generalize about even/odd degree polynomial and multiplicity. As

an example, following is a screen shot of a task where students had to produce the

graph and after analyzing the graph they had to identify the four zeros of the function

f(x) = 2x4-5x3-x2 using the concept of even function and multiplicity.

f(x)=2(x)^4-5(x)^3-(x)^2

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-12

-10

-8

-6

-4

-2

2

4

x

y

Figure 2: Screen shot of a fourth degree polynomial

Research Instruments:

In this research, students’ understanding was measured by post-test. The

post-test (See Appendix B) contained five questions, among which one question was

knowledge based, one question was comprehension based, one question was

application based (Plotting graph), and two questions were based on analytical thinking.

After the post-test a student survey was conducted. The survey had six questions and

each question had five answer choices. The survey was only conducted to the


experimental group to gather information regarding students’ opinion on the software

and its influence on the learning process. The questions of the survey were related to

the students’ level of motivation in using the software, helpfulness of the software in

learning the concepts, level of difficulty in using the software, students’ level of

confidence about the concepts, and students’ level of interests in future use of the

software. Another survey was also conducted at the beginning of the study to gather

information about participants. Through this survey the researcher collected information

regarding student demographics and their habit of computer use.

Procedures:

In this study the research design used was the Posttest only Experimental

design. The study took three weeks to complete. At the beginning of the study the

participants were grouped into two different groups (Control and Experimental) through

‘Matching’ process. The teacher who was delivering the instruction to both groups was

also trained during the first two days of the study. Both the control and experimental

groups included students from grades 8th, 9th and 10th. Each group was consist of

fifteen participants who are currently taking Algebra I. Both groups of students were

taught using lecture, classroom discussion, and text book readings and activities, but

only the experimental group had access to the graphing software. Participants of the

experimental group had access to eight desktop computers equipped with the graphing

software during the instruction. The groups were also given twelve instructional days to

learn the material and one day to prepare for the test. Both groups were given forty

minutes to take the post-test towards the end of the research study. During this study

the groups explored concepts related to Polynomial Functions (i.e. Graphing,


Translating, Transforming, Factor Theorem and Analyzing solutions) from the book

Advanced Algebra, published by Scott, Foresman.

Analysis:

In this study statistical analysis was done by using SPSS computer program. The

statistical analysis was included both descriptive statistics and inferential statistics. The

descriptive analysis was incorporated by mean and standard deviation of the data. The

inferential statistics included t-test to determine the significance of the difference

between the two different means. Scores from the post-test was compared using an

independent sample t-test of statistical significance. The t-test measured the difference

between the mean of the scores of the post-test from the control group (who did not use

graphing software during the instruction) and the experimental group (who used the

software to analyze polynomial functions). The student survey that was conducted to

the experimental group was also analyzed by using MS Excel.


FINDINGS

This chapter presents and analyzes the findings of the research project. The

findings will be described in two major sections. The first section will contain an analysis

of the post-test scores of the two groups (Control and Experimental) and the second

section will contain an analysis on the ‘Student Survey’ (See Appendix C) which was

conducted after the post-test. In this study the t-test analysis of the post-test scores was

not statistically significant (t(28)=-0.306, p>.05). Interestingly though, in two (analysis

base questions) out of the five questions in the post-test, students in the experimental

group showed significantly higher understanding of the concepts, and in one

(application base question) out of the five questions, students in the control group

showed significantly higher understanding of the concepts. After the post-test, the

researcher conducted a student survey with the experimental group in order to gather

information regarding students’ opinion on the software and its influence on their

learning process.

Findings from post-test:

A post-test was administered to both the experimental and control groups. All

thirty students from the two groups had participated in the test. The test had five

questions and among those questions one question was knowledge based, one

question was comprehension based, one question was application based, and two

questions were analysis based. Students were not allowed to use any calculators or the

computer software. It was a paper-pencil based test, where the students had to write

their response and show their work to receive full/partial credit. Each question was

worth 20 points, thus the total adding up to 100 points.


Table 1 below shows the analysis of the scores of the first question (see

Appendix B) in the post-test. This question was knowledge based and it required

students to use the definition of polynomial in order to find the values of given terms.

The difference between the means (%) of the control and experimental groups is less

than 2%. Students in the experimental group scored slightly better in this question, but

the t-test (t(28)=-0.235, p>.05) shows that the difference between the means (%) of the

two groups is not significant. This indicates that using the software did not provide any

significant advantage to the experimental group in the knowledge based question.

Table 1

Question 1: Knowledge BaseGroups N Mean (%) Std. Dev. t df Sig. (2-tailed)

1(Ctrl) 15 86.67 16.32

-0.235 28 0.816

2(Expr) 15 88.00 14.74

Table 2 below shows the analysis of the scores of the second question (see

Appendix B) in the post-test. This question was comprehension base and it asked the

students to identify whether the given functions were polynomials or not, and to state

the degrees and the leading coefficients of those functions. The difference between the

means of the scores of the control and experimental groups is less than 1%. Students in

the experimental group performed slightly better in this question, but the t-test (t(28)=-

0.314, p>.05) shows that the difference between the means of the two groups is not

significant. This indicates that using the software did not provide any significant

advantage to the experimental group in comprehension based questions.


Table 2

Question 2: Comprehension BaseGroups N Mean (%) Std. Dev. t df Sig. (2-tailed)

1(Ctrl) 15 89.33 10.33

-0.314 28 0.756

2(Expr) 15 90.67 12.80

Table 3 below shows the analysis of the scores of the third question (see

Appendix B) in the post-test. This question was application based and it asked the

students to find the equation of a fourth degree polynomial, and to sketch the graph of

that polynomial. The difference between the means of the control and experimental

groups is approximately 14%. Students in the control group scored higher in this

question, and the t-test (t(28)=3.28, p<.05) shows that the difference between the

means of the two groups is significant. This indicates that using the software did not

provide any significant advantage to the experimental group in application based

questions, and the control group was able to learn the application concepts better

without the use of the software. Since the students in the control group used paper-

pencil to graph during the instruction, it was expected that they would score higher than

the experimental group in this question.

Table 3

Question 3: Application BaseGroups N Mean (%) Std. Dev. t df Sig. (2-tailed)

1(Ctrl) 15 96.67 7.24

3.280 28 0.003


2(Expr) 15 82.67 14.86

Table 4 shows the analysis of the scores of the fourth question (see Appendix B)

in the post-test. This question was analysis base and the students were given a graph

of a polynomial function. After analyzing the graph students were supposed to find the

degree and the zeros of the function. They were also asked to find the equation of the

given graph. The difference between the means of the control and experimental groups

is approximately 8%. Students in the experimental group performed better in this

question, and the t-test (t(28)=-2.245, p<.05) shows that the difference between the

means of the two groups is significant. This indicates that using the software did provide

a significant advantage to the experimental group in analysis based questions.

Table 4

Question 4: Analysis BaseGroups N Mean (%) Std. Dev. t df Sig. (2-tailed)

1(Ctrl) 15 84.67 9.90

-2.245 28 0.033

2(Expr) 15 92.67 9.61

Table 5 below shows the analysis of the scores of the fifth question (see

Appendix B) in the post-test. This question was also analysis base and it tested

students’ understanding of translation and transformation, as well as the analysis of the

Factor Theorem. The difference between the means of the control and experimental

groups is approximately 7%. Students in the experimental group performed better in this

question, and the t-test (t(28)=-2.323, p<.05) shows that the difference between the

means of the two groups is significant. This indicates that using the software did provide


a significant advantage to the experimental group in analysis based questions.

Table 5

Question 5: Analysis BaseGroups N Mean (%) Std. Dev. t df Sig. (2-tailed)

1(Ctrl) 15 76.00 8.28

-2.323 28 0.028

2(Expr) 15 83.33 9.00

Table 6 below shows the analysis of the total scores of the five questions in the

post-test. The difference between the means of the control and experimental groups is

less than 1%. Students in the experimental group performed diminutively better in the

average scores in the post-test. Even though the t-test analysis (t(28)=-0.306, p>.05)

shows that the difference between the means (%) of the two groups is not statistically

significant, the analysis of the individual questions show that using graphing software

provided students with a greater achievement in analysis base questions.

Table 6

Questions 1 – 5: Total ScoresGroups N Mean (%) Std. Dev. t df Sig. (2-tailed)

1(Ctrl) 15 86.67 5.79

-3.06 28 0.762

2(Expr) 15 87.46 8.30

This finding concurs with Tall (1989), who examined the potential of well-

designed software to improve a learner’s concept image of function by allowing the

learner to explore the complex structures of functions. According to Tall (1989), the


software allows students to experience higher level cognitive structures than they would

be able to without such software. He suggested students can more readily explore

complex concepts through the use of technology.

Findings from student survey:

The student survey had six questions and each question had five answer

choices. The survey was conducted only to the experimental group (the group that used

graphing software during instruction). It was conducted to gather information regarding

students’ opinion on the software and its influence on the learning process.

The first question of the survey dealt with the students’ level of motivation when

learning the concepts using the software. Students could choose an answer from the

five choices provided in the survey. The analysis (Figure 3) of the responses shows that

60% of the students were above average motivated or highly motivated when the

software was integrated with the daily lessons.

Q1: How motivated were you to learn the concepts using the software?

0

2

4

6

3

01234567

Not motivatedat all

Sort ofmotivated

Averagemotivated

Above averagemotivated

Highlymotivated

Level of Motivation

Num

ber o

f Res

pons

es

Figure 3: Responses to question # 1

The second question of the survey was regarding to how helpful the software

was to the students when they used it during the lessons. Students could choose an


answer from the five choices provided in the survey. The analysis (Figure 4) of the

responses shows that thirteen out of fifteen (87%) of the students responded that the

software was above average helpful or highly helpful.

Q2: How helpful was the software in understanding the concepts?

01 1

76

012345678

Not helpful atall

Sort of helpful Average helpful

Above average helpful

Highly helpful

Num

ber o

f Res

pons

es


The third question dealt with the software’s level of difficulty when the students

were using it during the lessons. The analysis (Figure 5) of the responses shows that

almost half the students (47%) felt that the software had average level of difficulty when

using it during instructions. On the other hand, about 27% of the students felt the

software was above average or highly difficult, and the rest (about 27%) of the students

found that the software was sort of difficult or not difficult at all.


Q3: What is the level of difficulty in using the software?

2 2

7

3

1

012345678

Not difficult atall

Sort of difficult Averagedifficult

Above averagedifficult

Highly difficult

Num

ber o

f Res

pons

es


The fourth question of the survey dealt with the level of confidence of the

students in their learning of the concepts using the software. The analysis (Figure 6) of

the responses from the students shows that 60% of the students’ level of confidence

was above average or high. 27% of the students felt their confidence level was average,

and the rest of the students (about 13%) felt they were sort of confident.

Q4: How confident do you feel about the concepts that you have learned using the software?

0

2

4

7

2

012345678

Not confidentat all

Sort of confident

Average confident

Above average confident

Highly confident

Num

ber o

f Res

pons

es



The fifth question asked the students about their level of interest about the

concepts when using the software in their learning. The analysis (Figure 7) of the

responses shows that among the 15 students in the group, 8 students (About 53%)

responded that they had above average or high interest in the concepts. Only 27% of

the students had average interest and the rest of the 20% students had less than

average interest.

Q5: How interested were you in learning the concepts by using the software?

0

34

7

1

012345678

Not interestedat all

Sort of interested

Averageinterested

Above averageinterested

Highlyinterested

Num

ber o

f Res

pons

es


The last question of the survey was related to the future use of math software by

the teachers. The analysis (Figure 8) of the responses shows that 100% students

recommended that their future teachers should use math software during the

instructions. Among the students, 67% had above average or high recommendation for

using math software during instruction.


Q6: Would you recommend your future teacher to use software in teaching mathematics?

0 0

5

8

2

0123456789

Not at all Sort of Average Above average Highly

Num

ber o

f Res

pons

es


The analysis of the ‘student survey’ conducted with the experimental group

indicated that using software during the instruction improved students’ motivation,

confidence and interest. The high level of motivation among many of the students in the

experimental group was noticed by the researcher during the instruction. Students were

eager to share their understanding and experience with the class when they were

working with the software. During a guided discovery based activity in one of the lesson

involving ‘Translation and Transformation’, students spontaneously generated various

types of polynomial graphs using the software and share their results with their peer. In

her observation the researcher found that the students in the experimental group were

very confident about the contents and they were open about asking clarifying questions

to instructor. From the survey it is also evident that majority of the students were very

interested to use the software. Some of the students were highly fascinated and wanted

to get a copy of the software to use it at home.


CONCLUSION

The purpose of this research was to find whether or not using graphing software

during the instruction of polynomials can improve students’ understanding of function

and their ability to create and interpret graphical representation of functions. In this

study, using the graphing software, the experimental group investigated concepts

related to Polynomial Functions (i.e. Graph, Translation, Transformation, Factor

Theorem, and Analyzing Solutions). From the analysis of the findings it is evident that

the experimental group was confident in their understanding of function. They also

scored higher in questions that tested their ability to analyze and interpret graphical

representation of functions. The software helped the students to readily observe the

relationship among the functions, and they were able to make conjecture and test those

conjecture through graphical investigation. The software allowed students to spend

more time making conjectures and creating their own reasoning to either prove or refute

those conjectures. Rather than using time and mental effort to calculate values and

plotting graphs, through the use of this software students were able to focus more on

the overall behavior/pattern of different polynomial functions. All the activities that were

designed and implemented using the software facilitated students to gain higher

analytical thinking skills, and consequently they performed better on the questions that

required higher level thinking and analyzing.

In the conclusion of his research Simmt (1997) indicated that most of the

teachers used calculators to facilitate one or two guided discovery activities. They did

not use the calculators to facilitates and/or encourage the students to conjecture and

prove or refute ideas. In this research the students were not only conducted discovery


based activities, but also regularly encouraged to use the graphing software to

conjecture and discuss their results in class. From the findings of this research, it is

evident that these activities helped students to apply higher level thinking skills on those

questions in the post-test that required analysis and evaluation of graphs.

In his study Ruthven (1990) found that students developed three distinct

approaches to symbolizing a graph: an analytic-construction approach that exploits

mathematical knowledge, a graphic-trial approach that compares successive expression

graphs with the given graph, and a numeric-trial approach guided by the coordinates of

the given graph. My findings also confirm the results found by Ruthven (1990). In the

fourth and fifth questions of the post-test, students applied ‘graphic-trail’ approach

where they had to identify the zeros and the function of a given graph. The third

question required students to construct the graph and find its equation using the given

roots. Students applied ‘numeric-trail’ approach in their class activity which helped them

to answer this question. Moreover, an ‘analytical- construction approach that exploits

mathematical knowledge’ is a skill that students achieved through out the lessons which

incorporate the graphing software.

Interestingly, through this study the researcher observed that when graphing

tools were used during the instruction, a spontaneous shift in the teachers’ role enriched

the learning process. When paper-pencil was used to learn about graphing, the

teachers’ role was a task setter and an explainer. On the other hand, when students

used the software in their learning, teachers’ role shifted to a consultant, a facilitator and

a fellow investigator. Farrell (1996) also described that these shifts occurred within each

class as the teachers moved from not using the graphing calculator and computer


technology to the use of it; the shifts were also accompanied by a decrease in lecture

and an increase in group work.

It is also interesting that the statistical analysis of the students’ over all scores of

the post-test did not show a significant advantage in understanding the concepts of

polynomial functions when the students used graphing software during the instruction.

Two possible reasons for not attaining significant statistical differences are (1) the

sample size (total thirty participants) was too small to analyze the data and to get a

statistically significant result, (2) length of the treatment (total three weeks) was too

short to get useful information regarding students understanding using the software

during instructions. Therefore, it is recommended that future research needs to be

conducted with a significantly larger numbers of participants and over a longer period of

time.


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STUDENT SURVEY 1

Please choose ONE response from each of the following question ( from question # 1 to question # 5 ):

1. What is your current grade level?

[ ] 8th [ ] 9th [ ] 10th

2. What is your gender?

[ ] Female [ ] Male

3. Which race do you associate yourself the most?

[ ] African American [ ] Asian [ ] Caucasian [ ] Hispanic

4. Do you have access to a personal computer at your home?

[ ] Yes [ ] No

5. Do you have internet connection (dial-up, DSL, cable etc.) at your home?

[ ] Yes [ ] No

Please choose all that applies for the following question:

What activities do you perform on a regular basis (at least once a week) using your computer/internet?

[ ] Browse the internet for news

[ ] Browse the internet for homework help

[ ] Send and receive email

[ ] Send and receive text Messages

[ ] Do homework using computer

[ ] Word Processing tasks (Writing papers, essays etc.) using computer

- Thank you for your responses -


POST-TEST

1. Refer to the definition (in your math journal) of the nth degree polynomial and the polynomial 5x7 + 4x6 – 8x3 + 1.3x2 – x [5 X 4 = 20 points]

State the value of each of the following:

a. n b. an c. an-1 d. a0 e. a3

2. In a – d, tell whether or not the expression is a polynomial. If it is, state its degree and leading coefficient: [4 X 5 = 20 points]

a. 3x + 7 b. 7x2 + 13x – 8 c. 5x6 + 4x3 – 15 d. 2x3 + 3x

3. 6, 4, – 1 and – 3 are the four roots of a fourth degree polynomial. [2 X 10 = 20 points]

a. Write an equation of the polynomial

b. Sketch the graph of the polynomial in an X-Y Plane

4. Analyze the following graph. [2 X 10 = 20 points]

a. Find the degree of this polynomial function and indicate the zeros b. Write the equation of the graph


5. Graph of cubic function may have any one of the following four types of shapes below. [2 X 10 = 20 points]

Type 1 Type 2 Type 3 Type 4

Using the Factor Theorem find the equation for a cubic function

a. with three x-intercepts whose graph looks like ‘type 2’b. with one x-intercepts whose graph looks like ‘type 3’


STUDENT SURVEY 2 – EXPERIMENTAL GROUP

1. How motivated were you to learn the concepts using the software?

a. Highly motivatedb. Above average motivatedc. Average motivatedd. Sort of motivatede. Not motivated at all

2. How helpful was the software in understanding the concepts?

a. Highly helpfulb. Above average helpfulc. Average helpfuld. Sort of helpfule. Not helpful at all

3. What is the level of difficulty in using the software?

a. Highly difficultb. Above average difficultc. Average difficultd. Sort of difficulte. Not difficult at all

4. How confident are you about the concepts you have learned using the software?

a. Highly confidentb. Above average confidentc. Average confidentd. Sort of confidente. Not confident at all

5. How interested were you in learning the concepts by using the software?

a. Highly interestedb. Above average interestedc. Average interestedd. Sort of interestede. Not interested at all

Documents

Graphing Software in Understanding of Polynomial …sms35077/eed602/portfolio/research... · Web viewIn this study statistical analysis was done by using SPSS computer program. The